Zero divisors and units with small supports in group algebras of torsion-free groups |
| |
Authors: | Alireza Abdollahi Zahra Taheri |
| |
Institution: | 1. Department of Mathematics, University of Isfahan, Isfahan, Iran;2. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran |
| |
Abstract: | Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽2 is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. Exp. Math., 24 (2015), 326–338] to arbitrary fields. |
| |
Keywords: | Divisor graph group ring Kaplansky’s unit conjecture Kaplansky’s zero divisor conjecture torsion-free group zero divisor zero unit graph |
|
|